1.5 The plucking process

© M. Zollner & T. Zwicker 2002, 2020 Translation by Tilmann Zwicker

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1.5.3 String-bouncing

If a string is plucked with little force, it will approximately react as a linear system. This

means that doubling the initial displacement will also double the displacement at any instant

of the subsequent vibration process. Of course, any displacement is limited – at some point

the string will hit the frets on the fretboard. In doing so, it generates a somewhat rattling,

buzzing sound. To some degree, this is in fact a means of musical expression and thus not

something generally undesired.

In the book “E-Gitarren” by Day/Waldenmaier we find the recommendation: "A slight tilt of

the bridge makes it possible to adjust the action of the high E-string a little lower than that of

the low E-string. The latter has a more pronounced vibration amplitude and requires more

space that the high strings ". However, the transverse stiffness for all customary string sets is

higher for the low E-string (E2) than it is for the high E-string (E4) – why then would the

stiffer string require more space for its vibration? It is o.k. to concede this space to it; that

decision is, however, just as individual as the choice of the string diameter and cannot be

justified with a generally larger amplitude.

Fig. 1.31: String displaced at A (bold line), intermediate positions of the vibration (thin lines). In the left-hand

figure, the string was pressed to the guitar body and then released, on the right it was pulled up and released.

“Sattel” = nut; “Steg” = bridge.

The string is displaced in a triangular fashion by the plectrum (or the finger-tip, or –nail, or

teeth …). After the plucking process, the string moves in a parallelogram-like fashion – given

that we take a dispersion-free model as a basis (Fig. 1.31). However, this movement in the

shape of a parallelogram can only manifest itself if the string does not encounter any

obstacles. Frets are potential obstacles; their immediate vicinity has the effect that the string

does not only occasionally establish contact but hits them on a regular basis … with the

parallelogram-shaped movement being correspondingly changed. Fig. 1.32 shows (seen from

the side) a neck with the typical concave curvature. The axis-relations of this figure hold for

the following figures, as well.

Fig. 1.32: Fretboard geometry (strongly distorted due to the scale); lower surface of the resting string (dashed).

The frets are distorted into lines due to the strong magnification of the vertical dimension.

“Sattel” = nut, “Steg” = bridge; “Griffbrett” = fretboard.

1. Basics of the vibrations of strings

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If the string pressed down at point A (Fig. 1.33) has no contact to the frets, it can freely decay

in the dispersion-free model case. The string that has been lifted up, however, hits the 10th

fret

already after less than half the vibration period – its vibration-shape is completely destroyed.

Fig. 1.33: String-parallelogram. On the left, the string was pressed down and then released (uninhibited

vibration); on the right it was lifted up and then released (fret-bounce at the 10th

fret). “Griffbrett” = fretboard.

The well-versed guitarist will vary his/her “attack” as required and shape the sound of the

respective picked note via change of the picking-strength and –direction: both pressing-down

and lifting-up of a string happen. However, in particular when using light string sets, a further

vibration pattern occurs. It is generated as the string contacts the last fret (towards the bridge)

when being pressed down during plucking (Fig. 1.34). As soon as the string is released, a

transverse wave propagates in both directions and is first reflected at the last fret and then at

the bridge. Consequently, a peak running towards the nut is generated – it is reflected there

and bounces onto the first fret (right-hand part of the figure).

Fig. 1.34: String displacement at different points in time. On the left, the first half-period is shown, on the right

we see the subsequent process including bouncing off the first fret. Plucking happens at point A with contact to

the fretboard. The time-intervals are chosen such that the resolution is improved at first and after t = T/2.

Without dispersion. “Griffbrett” = fretboard.

Immediately the question pops up: how often does this case happen? Contact-measurement at

the last fret tells us: a lot. For better understanding, Fig. 1.35 depicts the connection between

plucking force (transverse force) and initial string displacement (at A). Since the transverse

forces often reach 5 N (or even 10 N occasionally), contact to the last fret often occurs.

Fig. 1.35: Connection between transverse force and string displacement, open string (left), string fretted at the

14th

fret (right), plucking point 14 cm (–––) and 6 cm (---) from the bridge. 2,1 mm clearance between the string

and the last fret (= 22nd

fret). B-string, 13 mil, calculations.

“Saitenauslenkung” = string displacement; “Querkraft” = transverse force.

1.5 The plucking process

© M. Zollner & T. Zwicker 2002, 2020 Translation by Tilmann Zwicker

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We can see from Fig. 1.35 that the string operates as a linear system only for soft plucking.

As soon as the string gets into contact with the last fret, the force/displacement characteristic

experiences a knee – a jump in the stiffness of the string occurs. This degressive characteristic

tends to correspond to the behavior of a compressor: despite stronger plucking force, the

string-displacement grows only moderately. However, here we also find a source of potential

misunderstanding, for displacement does not equal loudness! With the string establishing

contact to the last fret, the shape of the vibration deviates from the mentioned parallelogram,

and changes result in the spectrum, and thus in the sound.

For the following graphs, the E4-string of an Ovation guitar (EA-68) was plucked using a

plectrum; the electrical voltage of the piezo pickup built into the bridge was analyzed (i.e. the

force at the bridge). The location of plucking was at a distance of 125 mm from the bridge,

and the plectrum was pressed towards the guitar body such that a fretboard-normal vibration

was generated. Fig. 1.36 shows time function and spectrum for the linear case (no contract

between string and last fret). The voltage of the piezo jumps back and forth between 0 V and

0,4 V, with a duty cycle resulting from the division of the string (517:125, scale = 642 mm).

Given the transfer coefficient of 0,2 V/N (Chapter 6), the corresponding force at the bridge

calculates as 2 N, this representing good correspondence to Fig. 1.35. In this example, 2 N

forms the limit of linear operation – using a larger force makes the string bounce off the frets.

Fig. 1.36: Time-function and spectrum of the piezo-signal. The upper half of the left-hand graph shows the

measured time function, below is the result of the calculation. On the right is the measured spectrum and the

(idealized) envelope. Open E4-string, fretboard-normal vibration. “Frequenz” = frequency.

The analyses shown in the following graphs (Fig. 1.37) correspond to Fig. 1.36 but are based

on (fretboard-normal) string excitations of different strengths. For the upper two pairs of

graphs we can see proportionality in the time domain and in the spectral domain: the level

spectrum is simply shifted upwards for stronger plucking. As soon as the plucking force

exceeds 2 N (in the lower two pairs of graphs), the string touches the last fret and bounces off

it. Time function and spectrum become irregular. The strong peak in the time function finds

its counterpart in the location function (Fig. 1.34); it may be interpreted as the interaction

between two excitations:

a) string displacement, force step at t = 0 (idealized), and

b) opposite-phase force step at the last fret; occurring at the instant as the string leaves the last

fret (t ≈ 0,2 ms).

1. Basics of the vibrations of strings

Translation by Tilmann Zwicker © M. Zollner & Tilmann Zwicker, 2002 & 2020

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Fig. 1.37: Time-function and spectrum of the piezo-voltage. String plucked with different force. See text.

1.5 The plucking process

© M. Zollner & T. Zwicker 2002, 2020 Translation by Tilmann Zwicker

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A spectral analysis encompassing the whole of the auditory range is conducive for the

acoustic guitar, and the same holds for a piezo-pickup (Chapter 6). In Fig. 1.38, three of the

sounds from Fig. 1.37 are shown as third-octave spectra. On the left, we see the spectra of

strings plucked lightly and with medium strength, respectively – the system is still linear and

the spectra merely experience a parallel shift. Strong plucking (right figure) leads to a level-

increase merely in the middle and upper frequency range; below 1 kHz, there is even a

decrease in level. As other strings are played, or as the E4-string is fretted at other frets, this

effect tends to remain, but the spectral differences are specific to the individual case.

Fig. 1.38: Third-octave spectra, open E4-string, overlapping analysis of main- and auxiliary third-octave.

On the left, and for the dashed curve on the right, there is not yet any bouncing off the frets. Strong plucking

(solid line of the right) causes the string to touch the last fret and bounce off it. 1st and 2

nd harmonic actually

decrease in this process, while there is a strong increase in level at middle and high frequencies.

From this, we can deduce a compressor-like behavior in any guitar: for light plucking, the

string operates as a linear system, and slight changes in the picking strength lead (with good

approximation) to similar level changes in the whole frequency range. However, already at

medium picking strength, the string bounces off the frets – the lower the action and the lighter

the strings, the lower is the threshold to this occurring. Now, if filtering (due to magnetic

pickups) accentuates a specific frequency range, this compression is perceived with different

strength. Fender-typical single-coil pickups emphasize the range around 3 – 5 kHz. This will

lead to less perception of compression compared to humbuckers sporting resonance

frequencies around 2,5 kHz. This may not happen for all played notes, but it does happen in

the example shown in Fig. 1.38. So does a humbucker compress more strongly than a single-

coil? “Somehow”, yes – but not causally. The source of the compression is the string (in

conjunction wit the frets) that compresses in different ways in various frequency ranges.

Pickups and amplifiers make this different compression audible in different ways.

Here’s an opinion voiced in the Gitarre & Bass magazine (02/2000): "What happens when I,

for example, pick the low E-string first softly and then more and more strongly via a slightly

distorted amp? The Strat behaves much more dynamically and you can open the throttle ever

more until, purely theoretically, the string throws in the towel and breaks. The Les Paul shows

an entirely different character: first, the increasingly harder picking also generates more

loudness, but then the whole thing topples over: the notes don’t get louder anymore but more

dense – almost as if there were a compressor/limiter switched in. Say what?! Indeed, the

information of the string vibrations resulting from the behavior of the wood determines the

tonal characteristic of the Les Paul, but not the fatter sounding humbuckers.”

1. Basics of the vibrations of strings

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The G&B-author was careful (?) enough not to throw in something like “and that shows that

mahogany compresses more strongly than alder”. Still, he infers: “now we understand, why a

Strat even with Humbuckers can never turn into a Les Paul. You can at most make the tone

warmer and fatter, but the typical compression is out of reach.” Unfortunately, the author

does not report which experiments or models were the basis for his last conjecture.

Fig. 1.39: Third-octave spectrum, Stratocaster, neck-

pickup, E2-string (42mil) fretted at the 5th

fret. Plucked

from lightly to strongly. Distance between plectrum and

bridge: 13 cm. Clearance of the open E2-string to the

last fret: 2,3 mm.

As we can see from Fig. 1.39, a Stratocaster, too, compresses in the range of the low

partials. While the level-difference between light and very strong plucking is no less than 39

dB at 4 kHz, the fundamental changes only by 7 dB. Your typical Gibson Humbucker will

only transmit the spectrum of the low E-string up to about 2 kHz and therefore misses the

dynamic happening in the 4-kHz-range that a Fender pickup will still capture. However, in the

experiment reported in G&B, it is likely that behavior of the amplifiers was almost more

important: “via a slightly distorted amp”. There you go! The Gibson Humbucker will have

generated approximately double the voltage of the Fender single-coil. That makes the

amplifier participate in the signal compression: it will compress (or limit) the louder signal

(that of the Les Paul).

However, that does not mean that the compression is determined merely by the action on the

guitar, and by the amplifier. As the string bounces off the fret, a metal hits metal (at least on

the electric guitar). The result is a broad-band bouncing noise that extends to the upper limit

of the audible frequency range. String- and fret-materials are of particular significance in this

bouncing noise: pure-steel wound strings generate a more aggressive, treble-laden noise

compared to pure-nickel wound strings. Old string with their winding filled up by rust, grease,

etc, will sound duller than fresh strings. And the fret-wire that the string hits (that may in fact

be any fret in the course of the vibration) contributes, with its mechanical impedance, to the

bouncing noise, as well. A detailed analysis of the mechanical neck- and body- impedances

follows in Chapter 7; string/fret-contacts are analyzed in detail in Chapter 7.12.2.